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Free Field realization of the $\hat{\mathcal{D}}_q$ Algebra for the $η$-$ξ$ system, Integrals of Motion and Characters

We introduce a free field realization of the central extension of the Lie algebra $\mathcal{D}_q$ of difference operators on the circle in terms of the fermionic $η$-$ξ$ system. This realization admits a nontrivial Jordan block structure. We also review the free field realization of $\mathcal{W}_{1+\infty}$ algebra, and point out some relations beween its generators of weight zero and the local integrals of motion of Bazhanov Lukyanov and Zamolodchikov. Finally we compute the finitized characters, and the continuum characters of the Local Integrals of Motion, and find out and interesting analogy with the generating functions for the counting of branched covers of elliptic curves.